What positive integer y satisfies 3x^2+ y^2− 16 = 3(x^2+ 3)?
To simplify the equation, distribute the 3 on the right side of the equation to get 3x^2 + y^2 - 16 = 3x^2 + 9.
Subtract 3x^2 from both sides to get y^2 - 16 = 9.
Add 16 to both sides to get y^2 = 25.
Take the square root of both sides of the equation to get y = ±5.
Since we're looking for a positive integer solution, the positive integer that satisfies the equation is y = 5. Answer: \boxed{5}.